Algebra homomorphism: Difference between revisions

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Algebra over a field#Algebra homomorphisms

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+#Redirect [[Algebra over a field#Algebra homomorphisms]]
-{{Short description|Ring homomorphism preserving scalar multiplication}}
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-{{Being merged|spacetype=|associative algebra||section=|inactive=|date=March 2023|nocat=|dir=to}}
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-In [[mathematics]], an '''algebra homomorphism''' is a [[homomorphism]] between two [[algebra over a field| algebra]]s. More precisely, if ''A'' and ''B'' are algebras over a [[field (mathematics)|field]] (or a [[ring (mathematics)|ring]]) ''K'', it is a [[function (mathematics)|function]] {{nowrap|''F'' : ''A'' → ''B''}} such that, for all ''k'' in ''K'' and ''x'', ''y'' in ''A'', one has{{refn|{{cite book | last1=Dummit | first1=David S. | last2=Foote | first2=Richard M. | title=Abstract Algebra | publisher=[[John Wiley & Sons]] | year=2004 | edition=3rd | isbn=0-471-43334-9}}}}{{refn|{{cite book | last1=Lang | first1=Serge | author1-link=Serge Lang | title=Algebra | publisher=[[Springer Science+Business Media|Springer]] | series=[[Graduate Texts in Mathematics]] | year=2002 | isbn=0-387-95385-X}}}}
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-* <math>F(kx) = kF(x)</math>
+}}
-* <math>F(x + y) = F(x) + F(y)</math>
-* <math>F(xy) = F(x) F(y)</math>
-The first two conditions say that ''F'' is a ''K''-[[linear map]], and the last condition says that ''F'' preserves the algebra multiplication. So, if the algebras are [[associative algebra|associative]], ''F'' is a {{not a typo|[[rng homomorphism]]}}, and, if the algebras are rings and ''F'' preserves the [[multiplicative identity|identity]], it is a [[ring homomorphism]].
-If ''F'' admits an [[Inverse function|inverse]] homomorphism, or equivalently if it is [[bijective]], ''F'' is said to be an [[isomorphism]] between ''A'' and ''B''.
-== Unital algebra homomorphisms ==
-If ''A'' and ''B'' are two unital algebras, then an algebra homomorphism {{nowrap|''F'' : ''A'' → ''B''}} is said to be ''unital'' if it maps the unity of ''A'' to the unity of ''B''. Often the words "algebra homomorphism" are actually used to mean "unital algebra homomorphism", in which case non-unital algebra homomorphisms are excluded.
-A unital algebra homomorphism is a (unital) [[ring homomorphism]].
-== Examples ==
-* Every ring is a '''Z'''-algebra since there always exists a unique homomorphism {{nowrap|'''Z''' → ''R''}}. See ''{{slink|Associative algebra#Examples}}'' for the explanation.
-* Any homomorphism of commutative rings {{nowrap|''R'' → ''S''}} gives ''S'' the structure of a [[commutative algebra|commutative {{mvar|R}}-algebra]]. Conversely, if ''S'' is a commutative ''R''-algebra, the map {{nowrap|''r'' ↦ ''r'' ⋅ 1<sub>''S''</sub>}} is a homomorphism of commutative rings. It is straightforward to deduce that the [[overcategory]] of the commutative rings over ''R'' is the same as the category of commutative ''R''-algebras.
-* If ''A'' is a [[subalgebra]] of ''B'', then for every [[group of units|invertible]] ''b'' in ''B'' the function that takes every ''a'' in ''A'' to ''b''<sup>−1</sup> ''a'' ''b'' is an algebra homomorphism (in case {{nowrap|1=''A'' = ''B''}}, this is called an inner automorphism of ''B'').  If ''A'' is also [[simple algebra|simple]] and ''B'' is a [[central simple algebra]], then every homomorphism from ''A'' to ''B'' is given in this way by some ''b'' in ''B''; this is the [[Skolem–Noether theorem]].
-== See also ==
-* [[Morphism]]
-* {{slink|Universal algebra|Basic constructions}}
-* [[Spectrum of a ring]]
-* [[Augmentation (algebra)]]
-== References ==
-{{reflist}}
-{{DEFAULTSORT:Algebra Homomorphism}}
-[[Category:Algebras]]
-[[Category:Ring theory]]
-[[Category:Morphisms]]